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lec16 - Scott Hughes Massachusetts Institute of Technology...

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Scott Hughes 12 April 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 16: RL Circuits. Undriven RLC circuits; phasor representation. 16.1 RL circuits Now that we know all about inductance, it’s time to start thinking about how to use it in circuits. A simple circuit that illustrates the major concepts of inductive circuits is this: V L R S2 S1 Suppose that at t = 0, we close the switch S 1, leaving S 2 open. How does the current evolve in the circuit after this? Let’s first think about this physically. Lenz’s law tells us that the magnetic flux through the inductor does not “want” to change. Since this flux is initially zero, the inductor will impede any current that tries to flow it — the current will initially try to remain at zero. As time passes, current will gradually leak through, and the magnetic flux will build up. Eventually, it should saturate at I = V/R — at this point the most current possible is flowing through the circuit. 16.1.1 To Kirchhoff or not to Kirchhoff We’d now like to analyze this circuit quantitively as well. In all other circumstances in which we’ve examined a circuit, we’ve used Kirchhoff’s laws for this; for a a simple single- loop circuit, the relevant rule is Kirchhoff’s second law, “The sum of the EMFs and the voltage drops around a closed loops is zero”. This rule is essentially a restatement of the old electrostatics rule that H ~ E · d~s = 0. This rule DOES NOT HOLD when we have a time changing magnetic field! In other words, inductors completely invalidate — at least formally — the foundation of Kirchhoff’s second law. 146
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So what do we do? Let’s reconsider: in place of H ~ E · d~s = 0, we have Faraday’s law: I ~ E · d~s = - 1 c d Φ B dt = E ind = L dI dt . The closed loop integral of ~ E gives us the induced EMF. I personally find it best to think of the inductor as an EMF source; it acts in the circuit effectively like a battery would. Even though Kirchhoff’s laws no longer stand, formally, on a very solid foundation, we can extend the formalism by thinking of inductors in the circuit as a source of EMF. The equations then carry over just fine; indeed, they carry over so well that almost all textbooks tell you to just apply Kirchhoff’s laws to circuits containing inductors without even mentioning this subtlety. It’s always worth bearing in mind that what you are really using is Faraday’s law. I will try to refer to the loop rule as Faraday’s law; it’s such a habit to call it Kirchhoff’s law, though, that I will surely slip from time to time. The tricky thing in these kinds of problems is getting the signs right. In all of these cases, Lenz’s law should guide you: the induced EMF will act in such a way as to oppose changes in the inductor’s magnetic flux. If a circuit initially has no current flowing through it, the inductor will oppose the build-up of current; if it initially does have current flowing, the inductor will try to keep that current flowing. Keep this intuition in mind and you should have no problem choosing the correct sign for the inductor’s EMF.
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